<PRE><B>Question 86013</B>
A right triangle is formed in the problem you described.  The right triangle has as its 
hypotenuse, the length of the rope, and its two legs are: (1) the 14 ft distance from the base
of the pole to the point where the rope just touches the ground and (2) the height of the pole.
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You know the 14 ft leg, and you need to find the other leg that is opposite the 60 degree 
angle.  The tangent is the function to use. Use the definition of the tangent which is
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{{{tan(A) = (opp/adj)}}}
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where opp represents the side opposite the angle, and adj represents the side adjacent
to the 60 degree angle.
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In this case you can substitute the known values to get:
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{{{tan(60) = opp/14}}}
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Multiplying both sides of this equation by 14 results in:
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{{{opp = 14*tan(60) = 14*(1.732050808) = 24.24871131}}}
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[Note that you can get tan(60) from your calculator by setting it to degrees mode, 
entering 60, and pressing the &quot;tan&quot; key. If you do this correctly, your calculator 
will give you the answer of 1.732050808.]
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The answer of 24.24871121 indicates that the height of the flagpole (the side opposite the 
60 degree angle) is approximately 24.25 feet which is about 24 feet 3 inches.
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Another way you can do this is by recognizing that a 60 degree right triangle has sides
that are in the following proportions: the hypotenuse = 2, the side opposite the 60 degree
angle is {{{sqrt(3)}}} and the side adjacent to the 60 degree angle is 1. You can check
this with the Pythagorean theorem which will show that:
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{{{2^2 = (sqrt(3))^2 + 1^2 = 3 + 1}}}
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Since you are involved with the side opposite and the side adjacent you can establish 
the following proportion involving similar triangles:
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{{{(standard*side*opposite)/(standard*side*adjacent) = (flagpole)/14}}}
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But the standard side opposite is {{{sqrt(3)}}} and the standard side adjacent is 1. 
Substituting these into the proportion results in:
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{{{sqrt(3)/1 = (flagpole)/14}}}
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You can solve this by multiplying both sides by 14 to get rid of the denominator of the
term containing &quot;flagpole&quot;. When you do the equation becomes:
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{{{flagpole = 14*sqrt(3)}}}
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and if you use a calculator to multiply out the right side you again get:
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{{{flagpole = 24.24871131}}}
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which also indicates that the flagpole is about 24.25 feet long.
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Hope this helps you to understand the problem.</PRE>